Quantifying the Visual Mechanisms Underlying the Rod-and-Frame Illusion

Perception Task: RFI

To determine the change in PSE as a function of frame size, we subtracted the PSE for counterclockwise trials from clockwise trials, and then divided that by half.

We observed a significant effect for each of the frame sizes (all p-values<0.001).
A one-way ANOVA was conducted to evaluate if there was a difference in magnitude between frame sizes, which revealed a main effect of frame size (stats).
A Tukey post hoc analysis revealed a statistically significant difference the small frame and extra large frame, but no other differences were statistically significant.

Descriptives

(#tab:percept summary stat table) Perception Task
Frame Size	Mean	Median	SD	Min	Max
Small	2.11	2.00	1.15	-0.27	6.35
Medium	1.91	1.71	1.12	0.15	6.68
Large	1.71	1.45	1.02	-0.06	4.77
Extra Large	1.55	1.33	0.99	-0.50	5.80

T-Tests

Perception Task: RFI
frame_size	estimate	statistic	p	parameter	Method	Alternative	95% CI
175	2.11	16.44	< .001***	79.00	One Sample t-test	two.sided	[1.85, 2.37]
410	1.91	15.27	< .001***	79.00	One Sample t-test	two.sided	[1.66, 2.16]
645	1.71	14.97	< .001***	79.00	One Sample t-test	two.sided	[1.48, 1.93]
880	1.55	14.03	< .001***	79.00	One Sample t-test	two.sided	[1.33, 1.77]

Frame Size ANOVA

p_mag_anova<- aov(frame_effect_perception ~frame_size, data =perception)
summary(p_mag_anova)

##              Df Sum Sq Mean Sq F value Pr(>F)   
## frame_size    3   14.3   4.759   4.154 0.0066 **
## Residuals   316  362.0   1.146                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

tukey_p<-tukey_hsd(p_mag_anova)
tukey_p

## # A tibble: 6 × 9
##   term  group1 group2 null.value estimate conf.low conf.high  p.adj p.adj.signif
## * <chr> <chr>  <chr>       <dbl>    <dbl>    <dbl>     <dbl>  <dbl> <chr>       
## 1 fram… 175    410             0   -0.202   -0.639    0.235  0.631  ns          
## 2 fram… 175    645             0   -0.402   -0.839    0.0348 0.0837 ns          
## 3 fram… 175    880             0   -0.562   -0.999   -0.125  0.0055 **          
## 4 fram… 410    645             0   -0.200   -0.637    0.237  0.638  ns          
## 5 fram… 410    880             0   -0.360   -0.797    0.0770 0.146  ns          
## 6 fram… 645    880             0   -0.160   -0.597    0.277  0.781  ns

apa_table(tukey_p)

(#tab:perc aov frame size effect)
term	group1	group2	estimate	conf.low	conf.high	p.adj	p.adj.signif
frame_size	175	410	-0.20	-0.64	0.24	0.63	ns
frame_size	175	645	-0.40	-0.84	0.03	0.08	ns
frame_size	175	880	-0.56	-1.00	-0.12	0.01	**
frame_size	410	645	-0.20	-0.64	0.24	0.64	ns
frame_size	410	880	-0.36	-0.80	0.08	0.15	ns
frame_size	645	880	-0.16	-0.60	0.28	0.78	ns

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = frame_effect_perception ~ frame_size, data = perception)
## 
## $frame_size
##               diff        lwr         upr     p adj
## 410-175 -0.2019557 -0.6390396  0.23512813 0.6314914
## 645-175 -0.4022388 -0.8393226  0.03484511 0.0836585
## 880-175 -0.5620776 -0.9991615 -0.12499376 0.0055008
## 645-410 -0.2002830 -0.6373669  0.23680084 0.6377398
## 880-410 -0.3601219 -0.7972058  0.07696198 0.1464796
## 880-645 -0.1598389 -0.5969227  0.27724500 0.7808421

## $frame_size
##  175  410  645  880 
##  "a" "ab" "ab"  "b"

## # A tibble: 4 × 4
##   frame_size  mean quant cld_p
##   <fct>      <dbl> <dbl> <chr>
## 1 175         2.11  2.52 a    
## 2 410         1.91  2.18 ab   
## 3 645         1.71  2.36 ab   
## 4 880         1.55  2.15 b

Saccade-to-Vertical Task: Visuovestibular Effect

The effect of the frames was quantified by subtracting the mean errors for the counterclockwise-tilted frames from those of the clockwise-tilted frames then halving this value to get a measure of the average effect of a single frame.

We observed a significant effect for each of the frame sizes (all p-values<0.001).
A one-way ANOVA was conducted to evaluate if there was a difference in magnitude between frame sizes, which revealed no main effect of frame size (stats).

Descriptives

(#tab:s2v summary stat table) Saccade-to-Vertical Task
Frame Size	Mean	Median	SD	Min	Max
Small	1.47	1.31	1.63	-2.77	8.74
Medium	1.23	1.01	1.75	-1.84	8.34
Large	1.14	1.04	1.35	-1.63	5.91
Extra Large	1.04	0.88	1.51	-2.10	8.97

T-Tests

s2v_ttest_df<-saccade_to_vert_magnitude %>% group_by(FRAME_SIZE_VAL) %>% do(tidy(t.test(.$frame_effect_v)))
nice_table(s2v_ttest_df)

FRAME_SIZE_VAL	estimate	statistic	p	parameter	Method	Alternative	95% CI
175	1.47	8.04	< .001***	79.00	One Sample t-test	two.sided	[1.11, 1.83]
410	1.23	6.29	< .001***	79.00	One Sample t-test	two.sided	[0.84, 1.62]
645	1.14	7.55	< .001***	79.00	One Sample t-test	two.sided	[0.84, 1.44]
880	1.04	6.16	< .001***	79.00	One Sample t-test	two.sided	[0.70, 1.37]

Frame Size Anova

vv_mag_anova<- aov(frame_effect_v~FRAME_SIZE_VAL, data =saccade_to_vert_magnitude)
#summary(vv_mag_anova)
tukey_vv<-tukey_hsd(vv_mag_anova)
apa_table(tukey_vv)

(#tab:s2v aov frame size effect)
term	group1	group2	estimate	conf.low	conf.high	p.adj	p.adj.signif
FRAME_SIZE_VAL	175	410	-0.24	-0.88	0.40	0.77	ns
FRAME_SIZE_VAL	175	645	-0.33	-0.97	0.31	0.56	ns
FRAME_SIZE_VAL	175	880	-0.43	-1.07	0.21	0.31	ns
FRAME_SIZE_VAL	410	645	-0.09	-0.73	0.55	0.98	ns
FRAME_SIZE_VAL	410	880	-0.19	-0.83	0.45	0.87	ns
FRAME_SIZE_VAL	645	880	-0.10	-0.75	0.54	0.98	ns

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = frame_effect_v ~ FRAME_SIZE_VAL, data = saccade_to_vert_magnitude)
## 
## $FRAME_SIZE_VAL
##                diff        lwr       upr     p adj
## 410-175 -0.23837579 -0.8788647 0.4021131 0.7715415
## 645-175 -0.32587743 -0.9663663 0.3146114 0.5546178
## 880-175 -0.43057760 -1.0710665 0.2099113 0.3066241
## 645-410 -0.08750163 -0.7279905 0.5529872 0.9849084
## 880-410 -0.19220181 -0.8326907 0.4482871 0.8657073
## 880-645 -0.10470017 -0.7451890 0.5357887 0.9746602

## $FRAME_SIZE_VAL
## $FRAME_SIZE_VAL$Letters
## 175 410 645 880 
## "a" "a" "a" "a" 
## 
## $FRAME_SIZE_VAL$LetterMatrix
##        a
## 175 TRUE
## 410 TRUE
## 645 TRUE
## 880 TRUE

Correlations for each frame size

## `geom_smooth()` using formula = 'y ~ x'

Small Frame

#175
cor_s2v_perc_175

## 
##  Pearson's product-moment correlation
## 
## data:  s2v_perc_175$perception and s2v_perc_175$s2v
## t = -0.15077, df = 78, p-value = 0.8805
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.2359011  0.2034108
## sample estimates:
##         cor 
## -0.01706895

Medium Frame

#410
cor_s2v_perc_410

## 
##  Pearson's product-moment correlation
## 
## data:  s2v_perc_410$perception and s2v_perc_410$s2v
## t = 4.1293, df = 78, p-value = 9.057e-05
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.2247359 0.5884918
## sample estimates:
##       cor 
## 0.4235391

Large Frame

#645
cor_s2v_perc_645

## 
##  Pearson's product-moment correlation
## 
## data:  s2v_perc_645$perception and s2v_perc_645$s2v
## t = 3.7225, df = 78, p-value = 0.0003706
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.1844243 0.5603040
## sample estimates:
##       cor 
## 0.3884026

Extra Large Frame

#880
cor_s2v_perc_880

## 
##  Pearson's product-moment correlation
## 
## data:  s2v_perc_880$perception and s2v_perc_880$s2v
## t = 5.6445, df = 78, p-value = 2.569e-07
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.3615895 0.6780148
## sample estimates:
##       cor 
## 0.5385222

Visuovestibular models predicting perceptual effect for each frame size

Small frame

## 
## Call:
## lm(formula = vv ~ perc, data = df_small)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.1732 -0.9443 -0.1688  0.6118  7.2427 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.52049    0.38651   3.934  0.00018 ***
## perc        -0.02429    0.16113  -0.151  0.88055    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.644 on 78 degrees of freedom
## Multiple R-squared:  0.0002913,  Adjusted R-squared:  -0.01253 
## F-statistic: 0.02273 on 1 and 78 DF,  p-value: 0.8805

Medium frame

## 
## Call:
## lm(formula = vv ~ perc, data = df_medium)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.8382 -0.7158 -0.1747  0.7763  7.6688 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.03488    0.35465  -0.098    0.922    
## perc         0.66340    0.16066   4.129 9.06e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.595 on 78 degrees of freedom
## Multiple R-squared:  0.1794, Adjusted R-squared:  0.1689 
## F-statistic: 17.05 on 1 and 78 DF,  p-value: 9.057e-05

Large frame

## 
## Call:
## lm(formula = vv ~ perc, data = df_large)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.1883 -0.6084 -0.0252  0.6845  4.1134 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.2630     0.2750   0.956 0.341852    
## perc          0.5155     0.1385   3.723 0.000371 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.256 on 78 degrees of freedom
## Multiple R-squared:  0.1509, Adjusted R-squared:   0.14 
## F-statistic: 13.86 on 1 and 78 DF,  p-value: 0.0003706

Extra large frame

## 
## Call:
## lm(formula = vv ~ perc, data = df_xl)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.7192 -0.7819 -0.0548  0.7057  4.4342 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.2342     0.2670  -0.877    0.383    
## perc          0.8224     0.1457   5.644 2.57e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.278 on 78 degrees of freedom
## Multiple R-squared:   0.29,  Adjusted R-squared:  0.2809 
## F-statistic: 31.86 on 1 and 78 DF,  p-value: 2.569e-07

Saccade to Rod : Orientation Contrast Effect

The effect of the frames was quantified by subtracting the mean errors for the counterclockwise-tilted frames from those of the clockwise-tilted frames then halving this value to get a measure of the average effect of a single frame.

We observed a significant effect for each of the frame sizes (all p-values<0.001).
A one-way ANOVA was conducted to evaluate if there was a difference in magnitude between frame sizes, which revealed a main effect of frame size (stats).
A Tukey post hoc analysis revealed a statistically significant differences between the small frame and each of the other frame sizes.

Descriptives

(#tab:s2r summary stat table) Saccade-to-Rod Task: Orientation Contrast Effect
Frame Size	Mean	Median	SD	Min	Max
Small	1.61	1.50	1.44	-4.51	6.47
Medium	0.88	0.99	1.24	-2.39	4.38
Large	0.62	0.58	1.35	-4.02	4.01
Extra Large	0.76	0.70	1.14	-2.57	4.76

T-Test

FRAME_SIZE_VAL	estimate	statistic	p	parameter	Method	Alternative	95% CI
175	1.61	9.98	< .001***	79.00	One Sample t-test	two.sided	[1.29, 1.93]
410	0.88	6.36	< .001***	79.00	One Sample t-test	two.sided	[0.60, 1.15]
645	0.62	4.09	< .001***	79.00	One Sample t-test	two.sided	[0.32, 0.92]
880	0.76	5.94	< .001***	79.00	One Sample t-test	two.sided	[0.50, 1.01]

Frame Size Anova

##                 Df Sum Sq Mean Sq F value  Pr(>F)    
## FRAME_SIZE_VAL   3   46.9  15.635   9.284 6.7e-06 ***
## Residuals      316  532.2   1.684                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(#tab:s2r within graph)
term	group1	group2	estimate	conf.low	conf.high	p.adj	p.adj.signif
FRAME_SIZE_VAL	Small	Medium	0.73	0.20	1.26	0.00	**
FRAME_SIZE_VAL	Small	Large	0.99	0.46	1.52	0.00	****
FRAME_SIZE_VAL	Small	Extra Large	0.85	0.32	1.38	0.00	***
FRAME_SIZE_VAL	Medium	Large	0.26	-0.27	0.79	0.58	ns
FRAME_SIZE_VAL	Medium	Extra Large	0.12	-0.41	0.65	0.94	ns
FRAME_SIZE_VAL	Large	Extra Large	-0.14	-0.67	0.39	0.90	ns

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = frame_effect_r ~ FRAME_SIZE_VAL, data = saccade_to_rod_magnitude)
## 
## $FRAME_SIZE_VAL
##                          diff        lwr       upr     p adj
## Medium-Small        0.7300491  0.2000867 1.2600116 0.0024240
## Large-Small         0.9922527  0.4622903 1.5222152 0.0000123
## Extra Large-Small   0.8511250  0.3211626 1.3810875 0.0002514
## Large-Medium        0.2622036 -0.2677589 0.7921661 0.5777345
## Extra Large-Medium  0.1210759 -0.4088866 0.6510384 0.9350444
## Extra Large-Large  -0.1411277 -0.6710902 0.3888348 0.9017215

## $FRAME_SIZE_VAL
##       Large Extra Large      Medium       Small 
##         "a"         "a"         "a"         "b"

## # A tibble: 4 × 4
##   FRAME_SIZE_VAL   mean   quant cld_oc
##   <fct>           <dbl>   <dbl> <chr> 
## 1 Large          -0.618  0.0408 a     
## 2 Extra Large    -0.759 -0.151  a     
## 3 Medium         -0.880 -0.130  a     
## 4 Small          -1.61  -0.918  b

Correlations For Each Frame Size

Linear Models for Each Frame Size

Small Frame

oc_small<- lm(oc~perc, df_small)
oc_small_summary<- summary(oc_small)
oc_small_summary

## 
## Call:
## lm(formula = oc ~ perc, data = df_small)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.7726 -0.6195 -0.0129  0.6161  4.9510 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)   0.9789     0.3315   2.953  0.00416 **
## perc          0.2990     0.1382   2.163  0.03357 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.411 on 78 degrees of freedom
## Multiple R-squared:  0.05661,    Adjusted R-squared:  0.04451 
## F-statistic:  4.68 on 1 and 78 DF,  p-value: 0.03357

Medium Frame

## 
## Call:
## lm(formula = oc ~ perc, data = df_medium)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.0713 -0.6608  0.0778  0.6634  3.6698 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   0.2801     0.2653   1.056   0.2943  
## perc          0.3143     0.1202   2.615   0.0107 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.194 on 78 degrees of freedom
## Multiple R-squared:  0.08061,    Adjusted R-squared:  0.06882 
## F-statistic: 6.839 on 1 and 78 DF,  p-value: 0.0107

Large Frame

## 
## Call:
## lm(formula = oc ~ perc, data = df_large)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.4242 -0.5627 -0.0553  0.7771  2.9618 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.2068     0.2767  -0.747 0.457183    
## perc          0.4827     0.1393   3.465 0.000866 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.264 on 78 degrees of freedom
## Multiple R-squared:  0.1334, Adjusted R-squared:  0.1223 
## F-statistic:    12 on 1 and 78 DF,  p-value: 0.0008661

Extra Large Frame

## 
## Call:
## lm(formula = oc ~ perc, data = df_xl)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.4423 -0.5788 -0.0889  0.5603  3.4818 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   0.2485     0.2302   1.080   0.2837  
## perc          0.3296     0.1256   2.624   0.0105 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.102 on 78 degrees of freedom
## Multiple R-squared:  0.08111,    Adjusted R-squared:  0.06933 
## F-statistic: 6.885 on 1 and 78 DF,  p-value: 0.01045

Visuovestibular and Orientation Contrast Effect Correlations

Small Frame

## # A tibble: 80 × 5
## # Groups:   sid, frame_size [80]
##    sid         oc      vv frame_size   perc
##    <chr>    <dbl>   <dbl> <fct>       <dbl>
##  1 30201cc  2.68  -0.219  175         2.23 
##  2 30201ms  2.58   4.78   175         2.09 
##  3 30202is  0.463  1.76   175         3.99 
##  4 30205gr  1.98  -0.0316 175        -0.273
##  5 30205jp  1.52   0.700  175         0.685
##  6 30205sb -0.468  0.383  175         1.40 
##  7 30208lj  4.76   5.06   175         2.38 
##  8 30217af  2.10   0.389  175         2.23 
##  9 30301av  1.70   0.566  175         2.36 
## 10 30301sb  0.822  2.74   175         2.52 
## # ℹ 70 more rows

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between df_small$oc and df_small$vv is
## positive, statistically not significant, and tiny (r = 0.04, 95% CI [-0.19,
## 0.25], t(78) = 0.31, p = 0.756)

## `geom_smooth()` using formula = 'y ~ x'

report(small_vv_oc_cor)

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between df_small$oc and df_small$vv is
## positive, statistically not significant, and tiny (r = 0.04, 95% CI [-0.19,
## 0.25], t(78) = 0.31, p = 0.756)

Medium Frame

## # A tibble: 80 × 5
## # Groups:   sid, frame_size [80]
##    sid          oc     vv frame_size  perc
##    <chr>     <dbl>  <dbl> <fct>      <dbl>
##  1 30201cc  1.33   -1.19  410        1.72 
##  2 30201ms  2.03    4.13  410        3.54 
##  3 30202is  1.68    1.34  410        3.17 
##  4 30205gr -0.0156 -0.522 410        1.25 
##  5 30205jp  0.711   1.56  410        1.70 
##  6 30205sb -0.628   0.467 410        0.858
##  7 30208lj  2.31    3.76  410        1.62 
##  8 30217af  0.363  -0.368 410        0.991
##  9 30301av  1.62    1.46  410        2.29 
## 10 30301sb  0.0478  2.77  410        3.31 
## # ℹ 70 more rows

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between df_medium$oc and df_medium$vv
## is positive, statistically not significant, and very small (r = 0.08, 95% CI
## [-0.14, 0.29], t(78) = 0.70, p = 0.485)

## `geom_smooth()` using formula = 'y ~ x'

report(medium_vv_oc_cor)

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between df_medium$oc and df_medium$vv
## is positive, statistically not significant, and very small (r = 0.08, 95% CI
## [-0.14, 0.29], t(78) = 0.70, p = 0.485)

Large Frame

## # A tibble: 80 × 5
## # Groups:   sid, frame_size [80]
##    sid          oc     vv frame_size  perc
##    <chr>     <dbl>  <dbl> <fct>      <dbl>
##  1 30201cc  1.26   -0.916 645        1.54 
##  2 30201ms  2.97    1.17  645        4.11 
##  3 30202is  0.595   2.94  645        3.22 
##  4 30205gr  0.669   0.991 645        1.17 
##  5 30205jp  0.194   0.445 645        0.985
##  6 30205sb -0.918   1.04  645        0.749
##  7 30208lj  1.86    2.73  645        2.59 
##  8 30217af -0.0256  2.11  645        1.03 
##  9 30301av  0.656   1.58  645        3.30 
## 10 30301sb  1.27    2.87  645        4.03 
## # ℹ 70 more rows

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between df_large$oc and df_large$vv is
## positive, statistically not significant, and small (r = 0.20, 95% CI [-0.02,
## 0.40], t(78) = 1.78, p = 0.078)

## `geom_smooth()` using formula = 'y ~ x'

report(large_vv_oc_cor)

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between df_large$oc and df_large$vv is
## positive, statistically not significant, and small (r = 0.20, 95% CI [-0.02,
## 0.40], t(78) = 1.78, p = 0.078)

Extra Large Frame

## # A tibble: 80 × 5
## # Groups:   sid, frame_size [80]
##    sid          oc     vv frame_size  perc
##    <chr>     <dbl>  <dbl> <fct>      <dbl>
##  1 30201cc  1.20   -1.34  880        1.47 
##  2 30201ms  1.94    1.93  880        2.74 
##  3 30202is -0.876   1.89  880        2.58 
##  4 30205gr -0.0706  1.62  880        0.747
##  5 30205jp  0.872   0.113 880        0.285
##  6 30205sb  0.298   0.454 880        0.773
##  7 30208lj  0.957   1.91  880        1.30 
##  8 30217af -0.750   0.598 880        1.21 
##  9 30301av  0.818   1.07  880        2.77 
## 10 30301sb -0.645   0.130 880        3.30 
## # ℹ 70 more rows

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between df_medium$oc and df_medium$vv
## is positive, statistically not significant, and very small (r = 0.08, 95% CI
## [-0.14, 0.29], t(78) = 0.70, p = 0.485)

## `geom_smooth()` using formula = 'y ~ x'

report(xl_vv_oc_cor)

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between df_xl$oc and df_xl$vv is
## positive, statistically significant, and medium (r = 0.23, 95% CI [0.02, 0.43],
## t(78) = 2.13, p = 0.037)

Across Frames

## # A tibble: 320 × 5
## # Groups:   sid, frame_size [320]
##    sid        oc     vv frame_size  perc
##    <chr>   <dbl>  <dbl> <fct>      <dbl>
##  1 30201cc 2.68  -0.219 175         2.23
##  2 30201cc 1.33  -1.19  410         1.72
##  3 30201cc 1.26  -0.916 645         1.54
##  4 30201cc 1.20  -1.34  880         1.47
##  5 30201ms 2.58   4.78  175         2.09
##  6 30201ms 2.03   4.13  410         3.54
##  7 30201ms 2.97   1.17  645         4.11
##  8 30201ms 1.94   1.93  880         2.74
##  9 30202is 0.463  1.76  175         3.99
## 10 30202is 1.68   1.34  410         3.17
## # ℹ 310 more rows

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between all_task$oc and all_task$vv is
## positive, statistically significant, and small (r = 0.15, 95% CI [0.04, 0.25],
## t(318) = 2.63, p = 0.009)

## `geom_smooth()` using formula = 'y ~ x'

report(across_vv_oc_cor)

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between all_task$oc and all_task$vv is
## positive, statistically significant, and small (r = 0.15, 95% CI [0.04, 0.25],
## t(318) = 2.63, p = 0.009)

Combined Saccades and Perception Task Comparison

So far the perceptual and orientation contrast effect were reported as negative numbers, indicating that the perceptual response or saccade erred in the opposite direction of the tilt of the frame. However, it should be noted that for the purpose of making an additive comparison between summed saccade tasks and the perceptual response, we used the inverse value of the OC effect.

Descriptives

Frame Size Anova

Correlations

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between across_frames$perception and
## across_frames$combined_saccade is positive, statistically significant, and very
## large (r = 0.43, 95% CI [0.33, 0.51], t(318) = 8.45, p < .001)

## `geom_smooth()` using formula = 'y ~ x'

## 
##  Pearson's product-moment correlation
## 
## data:  comb_sacc_perc_175$perception and comb_sacc_perc_175$combined_saccade
## t = 1.2689, df = 78, p-value = 0.2083
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.08000506  0.35096232
## sample estimates:
##       cor 
## 0.1422117

## 
##  Pearson's product-moment correlation
## 
## data:  comb_sacc_perc_410$perception and comb_sacc_perc_410$combined_saccade
## t = 4.9876, df = 78, p-value = 3.619e-06
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.3049679 0.6420802
## sample estimates:
##       cor 
## 0.4917352

## 
##  Pearson's product-moment correlation
## 
## data:  comb_sacc_perc_645$perception and comb_sacc_perc_645$combined_saccade
## t = 4.9228, df = 78, p-value = 4.662e-06
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.2991548 0.6383051
## sample estimates:
##     cor 
## 0.48687

## 
##  Pearson's product-moment correlation
## 
## data:  comb_sacc_perc_880$perception and comb_sacc_perc_880$combined_saccade
## t = 5.7131, df = 78, p-value = 1.935e-07
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.3672545 0.6815287
## sample estimates:
##      cor 
## 0.543144

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between across_frames$perception and
## across_frames$combined_saccade is positive, statistically significant, and very
## large (r = 0.43, 95% CI [0.33, 0.51], t(318) = 8.45, p < .001)

## `geom_smooth()` using formula = 'y ~ x'

## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'

Perception and Saccade-to-Vertical

## `geom_smooth()` using formula = 'y ~ x'

Perception and Saccade-to-Rod

## `geom_smooth()` using formula = 'y ~ x'

## 
##  Pearson's product-moment correlation
## 
## data:  s2r_perc_175$perception and s2r_perc_175$s2r
## t = -2.1634, df = 78, p-value = 0.03357
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.43490358 -0.01920867
## sample estimates:
##        cor 
## -0.2379214

## 
##  Pearson's product-moment correlation
## 
## data:  s2r_perc_410$perception and s2r_perc_410$s2r
## t = -2.6151, df = 78, p-value = 0.0107
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.47405942 -0.06846902
## sample estimates:
##        cor 
## -0.2839148

## 
##  Pearson's product-moment correlation
## 
## data:  s2r_perc_645$perception and s2r_perc_645$s2r
## t = -3.4645, df = 78, p-value = 0.0008661
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.5414599 -0.1581624
## sample estimates:
##        cor 
## -0.3651887

## 
##  Pearson's product-moment correlation
## 
## data:  s2r_perc_880$perception and s2r_perc_880$s2r
## t = -2.6239, df = 78, p-value = 0.01045
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.47480242 -0.06942329
## sample estimates:
##        cor 
## -0.2847961

Task by Frame Size

#two way anova task X frame size summary stats

task_by_frame_long %>% 
  group_by(frame_size,task) %>% 
  get_summary_stats(magnitude,type = "mean_sd")

## # A tibble: 8 × 6
##   frame_size task  variable      n  mean    sd
##   <fct>      <fct> <fct>     <dbl> <dbl> <dbl>
## 1 175        vv    magnitude    80 1.47   1.63
## 2 175        oc    magnitude    80 1.61   1.44
## 3 410        vv    magnitude    80 1.23   1.75
## 4 410        oc    magnitude    80 0.88   1.24
## 5 645        vv    magnitude    80 1.14   1.35
## 6 645        oc    magnitude    80 0.618  1.35
## 7 880        vv    magnitude    80 1.04   1.51
## 8 880        oc    magnitude    80 0.759  1.14

#box plot 
task_frame_bxp<- ggboxplot(
  task_by_frame_long, x = "frame_size", y = "magnitude", color = "task", pallet ="jco")

task_frame_bxp

Model Comparison

For each frame size a hierarchical design was employed using two models: 1) model 1 predicted the overall RFI magnitude (measured by the perception task) from the visuovestibular effect (measured by the saccade-to-vertical task) and 2) model 2 predicted the overall RFI magnitude from visuovestibular effect and the orientation contrast effect (measured by the saccade-to-rod task).

Small Frame

Perceptual effect predicted by visuovestibular effect

## 
## Call:
## lm(formula = vv ~ perc, data = df_small)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.1732 -0.9443 -0.1688  0.6118  7.2427 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.52049    0.38651   3.934  0.00018 ***
## perc        -0.02429    0.16113  -0.151  0.88055    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.644 on 78 degrees of freedom
## Multiple R-squared:  0.0002913,  Adjusted R-squared:  -0.01253 
## F-statistic: 0.02273 on 1 and 78 DF,  p-value: 0.8805

Perceptual effect predicted by orientation contrast effect

## 
## Call:
## lm(formula = oc ~ perc, data = df_small)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.7726 -0.6195 -0.0129  0.6161  4.9510 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)   0.9789     0.3315   2.953  0.00416 **
## perc          0.2990     0.1382   2.163  0.03357 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.411 on 78 degrees of freedom
## Multiple R-squared:  0.05661,    Adjusted R-squared:  0.04451 
## F-statistic:  4.68 on 1 and 78 DF,  p-value: 0.03357

Model comparison

Model 1: perceptual effect~visuovestibular Model 2: perceptual ~ visuovestibular+ orientation contrast

## 
## Call:
## lm(formula = perc ~ vv + oc, data = df_small)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.4789 -0.6643 -0.1876  0.3187  4.2826 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.83034    0.21902   8.357 2.08e-12 ***
## vv          -0.01792    0.07779  -0.230   0.8184    
## oc           0.19003    0.08810   2.157   0.0341 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.129 on 77 degrees of freedom
## Multiple R-squared:  0.05726,    Adjusted R-squared:  0.03277 
## F-statistic: 2.338 on 2 and 77 DF,  p-value: 0.1033

## [1] 0.05696534

## Warning in anova.lmlist(object, ...): models with response '"perc"' removed
## because response differs from model 1

## Analysis of Variance Table
## 
## Response: vv
##           Df  Sum Sq Mean Sq F value Pr(>F)
## perc       1   0.061 0.06147  0.0227 0.8805
## Residuals 78 210.933 2.70427

tinytable_m5agzcoomjsclsrwkug8

	Model 1	Model 2
(Intercept)	1.520	1.830
	(0.387)	(0.219)
perc	-0.024
	(0.161)
vv		-0.018
		(0.078)
oc		0.190
		(0.088)
Num.Obs.	80	80
R2	0.000	0.057
R2 Adj.	-0.013	0.033
AIC	310.6	251.4
BIC	317.7	260.9
Log.Lik.	-152.296	-121.710
RMSE	1.62	1.11

Medium Frame

Perceptual effect predicted by visuovestibular effect

summary(model_1_medium)

## 
## Call:
## lm(formula = vv ~ perc, data = df_medium)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.8382 -0.7158 -0.1747  0.7763  7.6688 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.03488    0.35465  -0.098    0.922    
## perc         0.66340    0.16066   4.129 9.06e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.595 on 78 degrees of freedom
## Multiple R-squared:  0.1794, Adjusted R-squared:  0.1689 
## F-statistic: 17.05 on 1 and 78 DF,  p-value: 9.057e-05

Perceptual effect predicted by orientation contrast effect

## 
## Call:
## lm(formula = oc ~ perc, data = df_medium)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.0713 -0.6608  0.0778  0.6634  3.6698 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   0.2801     0.2653   1.056   0.2943  
## perc          0.3143     0.1202   2.615   0.0107 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.194 on 78 degrees of freedom
## Multiple R-squared:  0.08061,    Adjusted R-squared:  0.06882 
## F-statistic: 6.839 on 1 and 78 DF,  p-value: 0.0107

Model comparison Model 1: perceptual effect~visuovestibular Model 2: perceptual ~ visuovestibular+ orientation contrast

## 
## Call:
## lm(formula = perc ~ vv + oc, data = df_medium)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.92902 -0.56106 -0.08099  0.47815  3.13629 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.39053    0.15335   9.068 8.87e-14 ***
## vv           0.25768    0.06352   4.056 0.000118 ***
## oc           0.22763    0.08988   2.533 0.013356 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.985 on 77 degrees of freedom
## Multiple R-squared:  0.2425, Adjusted R-squared:  0.2228 
## F-statistic: 12.32 on 2 and 77 DF,  p-value: 2.273e-05

## [1] 0.0631005

## Warning in anova.lmlist(object, ...): models with response '"perc"' removed
## because response differs from model 1

## Analysis of Variance Table
## 
## Response: vv
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## perc       1  43.40  43.400  17.051 9.057e-05 ***
## Residuals 78 198.54   2.545                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

tinytable_lv2iw3c04ymsyz8yyjll

	Model 1	Model 2
(Intercept)	-0.035	1.391
	(0.355)	(0.153)
perc	0.663
	(0.161)
vv		0.258
		(0.064)
oc		0.228
		(0.090)
Num.Obs.	80	80
R2	0.179	0.242
R2 Adj.	0.169	0.223
AIC	305.7	229.5
BIC	312.9	239.1
Log.Lik.	-149.873	-110.775
RMSE	1.58	0.97

Large Frame

Perceptual effect predicted by visuovestibular effect

## 
## Call:
## lm(formula = vv ~ perc, data = df_large)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.1883 -0.6084 -0.0252  0.6845  4.1134 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.2630     0.2750   0.956 0.341852    
## perc          0.5155     0.1385   3.723 0.000371 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.256 on 78 degrees of freedom
## Multiple R-squared:  0.1509, Adjusted R-squared:   0.14 
## F-statistic: 13.86 on 1 and 78 DF,  p-value: 0.0003706

Perceptual effect predicted by orientation contrast effect

## 
## Call:
## lm(formula = oc ~ perc, data = df_large)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.4242 -0.5627 -0.0553  0.7771  2.9618 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.2068     0.2767  -0.747 0.457183    
## perc          0.4827     0.1393   3.465 0.000866 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.264 on 78 degrees of freedom
## Multiple R-squared:  0.1334, Adjusted R-squared:  0.1223 
## F-statistic:    12 on 1 and 78 DF,  p-value: 0.0008661

Model comparison Model 1: perceptual effect~visuovestibular Model 2: perceptual ~ visuovestibular+ orientation contrast

## 
## Call:
## lm(formula = perc ~ vv + oc, data = df_large)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.6401 -0.5876 -0.2332  0.2946  3.9705 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.28410    0.13581   9.455 1.59e-14 ***
## vv           0.24787    0.07650   3.240  0.00177 ** 
## oc           0.22699    0.07681   2.955  0.00415 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9027 on 77 degrees of freedom
## Multiple R-squared:  0.2373, Adjusted R-squared:  0.2175 
## F-statistic: 11.98 on 2 and 77 DF,  p-value: 2.948e-05

## [1] 0.08649254

## Warning in anova.lmlist(object, ...): models with response '"perc"' removed
## because response differs from model 1

## Analysis of Variance Table
## 
## Response: vv
##           Df  Sum Sq Mean Sq F value    Pr(>F)    
## perc       1  21.864 21.8644  13.857 0.0003706 ***
## Residuals 78 123.070  1.5778                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

tinytable_knxj2jxetcfztsrc6h03

	Model 1	Model 2
(Intercept)	0.263	1.284
	(0.275)	(0.136)
perc	0.516
	(0.138)
vv		0.248
		(0.076)
oc		0.227
		(0.077)
Num.Obs.	80	80
R2	0.151	0.237
R2 Adj.	0.140	0.218
AIC	267.5	215.6
BIC	274.6	225.1
Log.Lik.	-130.744	-103.798
RMSE	1.24	0.89

Extra Large Frame

Perceptual effect predicted by visuovestibular effect

## 
## Call:
## lm(formula = vv ~ perc, data = df_xl)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.7192 -0.7819 -0.0548  0.7057  4.4342 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.2342     0.2670  -0.877    0.383    
## perc          0.8224     0.1457   5.644 2.57e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.278 on 78 degrees of freedom
## Multiple R-squared:   0.29,  Adjusted R-squared:  0.2809 
## F-statistic: 31.86 on 1 and 78 DF,  p-value: 2.569e-07

Perceptual effect predicted by orientation contrast effect

## 
## Call:
## lm(formula = oc ~ perc, data = df_xl)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.4423 -0.5788 -0.0889  0.5603  3.4818 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   0.2485     0.2302   1.080   0.2837  
## perc          0.3296     0.1256   2.624   0.0105 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.102 on 78 degrees of freedom
## Multiple R-squared:  0.08111,    Adjusted R-squared:  0.06933 
## F-statistic: 6.885 on 1 and 78 DF,  p-value: 0.01045

Model comparison Model 1: perceptual effect~visuovestibular Model 2: perceptual ~ visuovestibular+ orientation contrast

## 
## Call:
## lm(formula = perc ~ vv + oc, data = df_xl)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.73611 -0.57671  0.00541  0.53652  2.24999 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.09815    0.12229   8.980 1.31e-13 ***
## vv           0.32691    0.06345   5.152 1.93e-06 ***
## oc           0.14516    0.08372   1.734   0.0869 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8264 on 77 degrees of freedom
## Multiple R-squared:  0.3167, Adjusted R-squared:  0.2989 
## F-statistic: 17.84 on 2 and 77 DF,  p-value: 4.294e-07

## [1] 0.02668074

## Warning in anova.lmlist(object, ...): models with response '"perc"' removed
## because response differs from model 1

## Analysis of Variance Table
## 
## Response: vv
##           Df  Sum Sq Mean Sq F value    Pr(>F)    
## perc       1  52.044  52.044   31.86 2.569e-07 ***
## Residuals 78 127.415   1.634                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

tinytable_lvmnbr7mch1b9m57mctx

	Model 1	Model 2
(Intercept)	-0.234	1.098
	(0.267)	(0.122)
perc	0.822
	(0.146)
vv		0.327
		(0.063)
oc		0.145
		(0.084)
Num.Obs.	80	80
R2	0.290	0.317
R2 Adj.	0.281	0.299
AIC	270.3	201.5
BIC	277.4	211.0
Log.Lik.	-132.132	-96.732
RMSE	1.26	0.81

Quantifying the Visual Mechanisms Underlying the Rod-and-Frame Illusion

Ernestine Brannon

Analysis

Perception Task: RFI

Descriptives

T-Tests

Frame Size ANOVA

Saccade-to-Vertical Task: Visuovestibular Effect

Descriptives

T-Tests

Frame Size Anova

Correlations for each frame size

Visuovestibular models predicting perceptual effect for each frame size

Saccade to Rod : Orientation Contrast Effect

Descriptives

T-Test

Frame Size Anova

Correlations For Each Frame Size

Linear Models for Each Frame Size

Visuovestibular and Orientation Contrast Effect Correlations

Combined Saccades and Perception Task Comparison

Descriptives

Frame Size Anova

Correlations

Task by Frame Size

Model Comparison

Small Frame

Medium Frame

Large Frame

Extra Large Frame